Let $\Lambda$ be a semisimple Artinian ring. Then $\Lambda=\bigoplus_{i=1}^r\Lambda_i$, ${} \Lambda_i\cong \text{Mat}{n_i}(D_i)$, ${} D_i$ is a division ring, and the $\Lambda_i$ are uniquely determined. The ring $\Lambda$ has exactly $r$ isomorphism classes of irreducible modules $M_i$, $i=1,\dots,r$, ${} \text{End}{\Lambda}(M_i)\cong D_i^{\text{op}} {}$, and $\text{dim}{\Delta_i^{\text{op} } }=n_i$. If $\Lambda$ is simple then $\Lambda\cong \text{Mat}{n}(D)$.
An inverse semigroup is a set $S$ with an associative binary operation such that for every $s\in S$ there exists a unique $s^{-1}\in S$ satisfying $ss^{-1}s=s$ and $s^{-1}ss^{-1}=s$.
Proposition: Let $S$ be an inverse semigroup. The set of idempotents $E(S) = {e\in S \mid ee=e}$ is a commutative inverse (sub)semigroup.
Corollary: Let $S$ be an inverse semigroup and $x,y\in S$. Then $(xy)^{-1}=y^{-1}x^{-1}$.
or take $\tr\mathcal{T}^N$
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${+,-}\to\set{1,\dots,q}$ (giving the $q$-state Potts model)
- action of $D_{2N}\times S_q$
- more interesting $\widetilde{H}$ (especially for $q\geq 3$)
- square $\to$ triangular/hexagonal etc.
- 2 dimensions $\to n$ dimensions
${} (2^{100} = 1267650600228229401496703205376) {}$